Bastian

In the default configuration, the Docker Image is virtual storage storing all Images, Containers and Volumes. If a container is miss-configured or not really well written, it could happen that it write more data in than the virtual storage is able to store. In that case you get "No Space left on Device" exception. Or there are just too many running container, that could also happen.
 
The first case can be check via the Container Size Button on the Docker Page. It lists all container by file system size. Someone else has to chime in, but I believe that excludes mount-bind paths.
 
If all your containers are running, you could try a docker system prune -af --volumes. This deletes all unused volumes, pending images und stopped containers. Note: Run it at your own risk!
 
To increase the size of the virtual disk, you have to stop your array, head over to Settings -> Docker and increase Docker vDisk size. This will delete all your Docker Data, so make sure everything important is (as it should be) mount-bind into the host system.

Sounds more like a configuration issue on your part. From that list I only know Plex and that should be able to survive a re-creation. Sometime the online service gets confused and you have to pin the libraries anew, but not the entire setup. Best to check your mount-bind against the official documentation. You can easily test it by just deleting the service. After creation from the template it should be like before.

You have to differentiate on which system your shells are running. User Scripts run in Unraids shell, which doesn't contain any rclone binary. If you open a Terminal on the Docker Tab, you actually start a session within the Container.
 
To replicate that behaviour you have to actually execute the command within the Container. The responsible command is called docker exec.
 
A possible User Scripts could look like
docker exec $(docker ps --filter=name=<NAME> -q) /bin/bash -c "YOUR COMMAND"  

What. The actual. F! This little comment on the bug report had the solution and I don't know how that could be missed. The global setting for NFS doesn't get regarded. I once used NFS, but disabled it globally after I had no use for it anymore. 
 
Enabled NFS -> Changed all NFS Export Settings -> (Disabled NFS again) -> Restarted the array and the symlinks are working now.

As you already said, /var/log is mostly logging of the host system. There you can find the log of unraid itself (syslog), the log of the docker engine (I presume, docker.log) and the log of various tools running on the host.
 
As stupid as it sounds, there is none. What you see in `docker logs` (or the UI) is what you get. You have to take into consideration that docker is no normalised ecosystem. It us up to each application and image maintainer how much and what they want to log. There are ways to extend the applications logging. For example, most server (Spring, Asp, etc) have different log levels, controlled through environment variables, but that depends very heavy on the exact implementation.
 
So, what to do if a container fails?
Does it fail on creation (unraid will prompt a "command failed" on saving)? You should get a pretty telling exception message from docker with the reason.
Does it exit with a non-zero code after creation? Check the docker logs. If the creator printed something, it should be in there. Otherwise you have to contact the creator (through an application thread here in the forum, github issue, etc) and attach both the docker log and docker run command (as printed by unraid on saving). They wrote the application and might see more than a third-party can.
 
There could be additional resources, like an internal log file, a configuration file created by the application, etc, but that varies from application to application. 
 
I just noticed that the UI only shows the last 100 or so lines. For the full log you can check the actual log file handled by docker. To get the path just run `docker inspect --format='{{.LogPath}}' <container id>`.
 
I hope it helped and could clear up your question somewhat